Notes on

The Man Who Changed Everything: The Life of James Clerk Maxwell, by Basil Mahon. Wiley 2004

Mathematicians such as the French Simeon-Denis Poisson, Andre-Marie Ampere, and the German Wilhelm Weber, had derived their electro and magnetic equations on the assumption that electrical charges and magnetic poles act on each other at a distance, while Faraday believed that electrical charges and magnets interact through lines of force which fill all space. To Maxwell, Faraday’s  idea of lines of force  rang true, but only needed a mathematical  expression. Maxwell discovered that incompressible fluid flow was an accurate analogy to the behavior of electrical charges and magnets, providing the mathematical framework for the lines of force concept.  Thus, the equations for describing incompressible fluid flow are analogs for electricity and magnetism, with pressure difference corresponding to potential difference, or voltage, and flow velocity corresponding to strength of electric or magnetic field. The new equations even  accounted for some electrical and magnetic effects which occurred at the boundary between different materials, but which could not be explained by action at a distance. Maxwell thus vindicated Faraday  and turned his ‘vague an varying ‘ lines of force into a new and mathematically impeccable concept, the field.[i]

Maxwell next set out to find a way to express the electrotonic state mathematically. With the aid of work by George Green, William Thompson, and George Gabriel Stokes, Maxwell used differential vectors (at a single point) and found that one of the vectors matched exactly Faraday’s concept of the electrotonic state. Initially he could not think of how to interpret the symbols physically. [ii]

The fluid flow analog only worked for static electric and magnetic fields and constant current. As soon as anything changed, the fields behavior was unlike any physical process known at that time. [iii] He modeled he dynamic behavior as spinning cells. He developed a mechanical analog for Faraday’s electrotonic state: it was the effect at any point in the field of the angular momentum Of the spinning cells. Like a flywheel, the cells would act as a store of energy, reacting with a counterforce to resist any change in their rotation. This takes the form of an electromotive force which would drive a current. [iv]

The next iteration in his theory came when he added elasticity to the cells/spheres. The softer the spring, the greater the electrical displacement. For a given potential difference. Electrostatic energy was potential energy; like a spring; magnetic energy was rotational, like a flywheel, and both could exist in empty space. A change in one always resulted in a change in the other. This new “elastic” model predicted two new phenomenon: 1) a new type of current, which would arise whenever the electric field is changes (displacement current). 2) all materials that have elasticity transmit waves. The EM waves were clearly transverse because the changing electric and magnetic fields were both at right angles to the direction of the wave. He compared his calculated EM wave velocity and compared it to the experimentally  measured speed of light. They  were very close. The physicists of the day believed a medium, an “aether” of some sort,  pervading all space,  was necessary to transmit light waves, so Maxwell’s theory fit right in. [v]

In his next iteration he suspected that the ultimate mechanisms of nature might be beyond our comprehension, so he set his model aside and chose to apply Lagrange’s method of treating the system like a black box: If you know the inputs and the systems general characteristics, you can calculate the outputs without knowledge of the internal mechanism. His first assumption is that  EM fields hold energy, both kinetic and potential. Electromotive and magnetomotive forces not forces in mechanical sense, but act in an analogous way.

Most of the quantities were vectors: The five main vectors were electric and magnetic field intensities, which resembled forces, electric and magnetic flux densities, which resembled strains, and electric current density, which was a kind of flow. Electric charge density  is a scalar. These were the fundamental quantities.  Everything came together beautifully. He showed that all aspects of the behavior of EM systems, including the propagation of light, could be derived from the laws of dynamics.

The theory is embodied in four equations in the 6 major unknowns. For a point in empty space:

Gaussian units:

Div E=0 implies no electric charge is present

Div H=0 implies that no single magnetic poles are present; they always come in north south pairs.

Curl E= -(1/c) dH/dt when the magnetic force changes, it wraps a circular electric force around itself

Curl H= (1/c) dE/dt when the electric force changes, it wraps s circular magnetic force around itself.

For a point in empty space, EM units:

Div E=0 implies no electric charge is present

Div H=0 implies that no single magnetic poles are present; they always come in north south pairs.

Curl E= - dH/dt when the magnetic force changes, it wraps a circular electric force around itself

Curl H= (1/c squared) dE/dt when the electric force changes, it wraps s circular magnetic force around itself.


E is the electric force and H the magnetic force at the arbitrary  point. C is the speed of propagation of the waves. It is also the ratio of the electro of the electromagnetic and electrostatic units of charge. One crucial assumption: that  electric currents exist in empty space. It is these “displacement currents” that give the equations their symmetry and make the waves possible. Without them, dE/dt =0 and the equations collapse. The keystone of his theory, the displacement current, had its origin in the idea that the spinning cells in his construction model could be springy. Heinrich Hertz produced and detected the EM waves predicted by Maxwell’s theory 20 years later.

Div and Curl are ways to describe how these forces vary in the space around the point. + Div is a measure of the tendency to be more outwards than inwards; Curl is a measure of  the force to curl or loop around the point.  [vi]

Book endnote:

For compactness,  modern vector notation is used. The substitution is legitimate because Maxwell himself later began the process of modernization. It is usual to include extra vectors B and D. The equations then become:


Div D=0 implies no electric charge is present

Div B=0 implies that no single magnetic poles are present; they always come in north south pairs.

Curl E= -(1/c) dB/dt when the magnetic force changes, it wraps a circular electric force around itself

Curl H= (1/c) dH/dt when the electric force changes, it wraps s circular magnetic force around itself.

D is the density of the electric flux produced by the electric field intensity E.

B is the density of the magnetic flux produced by the magnetic field intensity H.


But in Gaussian units, in empty space, D=E and B=H, allowing the equations to be written using E and H only.


E and H are not forces, strictly speaking, but rather the intensities of the E and H fields. They may be thought of as forces waiting to act, or that would be exerted upon a unit charge or unit magnetic pole if it were placed at that point.

In the Dynamical Theory Paper,  Maxwell expressed his results in more expansive form. He gave 8 equations for the electric and magnetic fields, remarking that “to eliminate a quantity which expresses a useful idea would be a loss rather than a gain at this stage of our inquiry.” One of the ideas thus expressed was Faraday’s electrotonic state, which in Maxwell’s scheme, became the momentum of the field. Oliver Heaviside  condensed the results to the abbreviated modern form.

Inconsistency in algebraic sign: More detail in Thomas K. Simpson ‘s guided study Maxwell on the Electromagnetic Field. Maxwell generally favored primacy of the field, as Farady had done, but some preferred to choose charge as the fundemental entity, and their case was strengthened when the electron was discovered in 1897. See Daniel M. Siegel Innovation in Maxwell’s Electromagnetic Theory.

There is an interesting short note in the Dynamical Theory paper about gravitation. It was natural for Maxwell to see whether his idea that energy existed in empty space could somehow explain gravity. He soon found that it could not.

End End Note Reference  [vii]


“James had so far written each relation involving vectors as a triple set; one for each x, y, and z direction.”

Quaternions invented by Sir William Rowan Hamilton and have four components: a scalar part, and a vector part in each of the x, y, z directions. Says quaternions were precursor to modern vector system, which dispenses with the scalar. Maxwell found that in some of his equations, 9 ordinary symbols could be replaced by 2 quaternion terms. He also found they made the physical meaning of the equations clearer. He included the shorthand quaternion notation in his Treatise on Electricity and Magnetism, along the conventional longhand notation. To help with the physical interpretation, he coined the terms “curl”,  convergence”, and “gradient” in his quaternion representation. Today “curl”,  convergence”, and “gradient” are used.  Gradient (grad)  represents the direction and rate of change of a scaler quantity in space. “James had started the process that gives us today the system of modern vector analysis. The job was completed around 20 years later by American Josiah Willard Gibbs and the Englishman Oliver Heaviside. [viii]

Accepted a post at Cambridge, where he built, with architect  WM Fawcett, the Cavendish, a lab. It became the birthplace for much of modern physics, and served Cambridge for 100 years. The electron was discovered at the Cavendish lab by  a successor of Maxwell there, JJ Thomson. [ix]

Maxwell’s Treatise is probably, after Newton’s Principia, the most renown book in the history of physics. The Treatise takes you along to a certain point, then, when you expect to go on, takes you back to the startand along a new path, which leads to areas inaccessible from the first. The book is not an atlas but an explorers report. In the Treatise, Maxwell made an important new prediction: that EM waves exert a radiation pressure.  He calculated that bright sunlight  presses on the earth’s surface with a force of four pounds per square mile.  In 1900, the Russian physicist, Pyotr Lebsdev experimentally confirmed Maxwell’s prediction. [x]

Maxwell and Boltzman’s theory on the behavior of gasses disagreed with measured values. The explanation  came 50 years later in quantum theory. [xi]

Hermann Helmholtz took Maxwell’s EM theory seriously. He was professor of physics at Berlin, and on the same level as Lord Kelvin in Britain. In 1879 he persuaded the Berlin Academy of Sciences to offer a prize for a conclusive experimental test of Maxwell’s theory. His star student, Heinrich Hertz took the challenge. He had to produce and detect either displacement currents,  or electromagnetic waves from an electrical circuit. Hertz finally succeded by making use of  a well known property of traveling waves: if they are reflected back upon the source, they form standing waves. Using a spark gap circuit he measured a wavelength of about 30 cm. Since  Hertz first detected the waves, they became known as “hertzian” waves.[xii]

Does not even mention Tesla.

Einstein brooded over an apparent conflict between Maxwell’s equations and Newton’s laws of motion. It stemmed from the Michaelson Morley experiment which suggested that light always travels at the same speed, no matter how fast or in what direction the observer is moving. This could be explained if distances and times appeared different to observers traveling at different speeds. A formula for converting the times and distances measured by one observer to those measured by another (traveling at different speeds) was put forward by  Hendrik Antoon Lorentz. Maxwell’s equations worked perfectly under this transformation, but Newton’s did not. Einstein solved the problem with his special theory of relativity. He took the constancy of the speed light as a starting point and worked out the consequences. He arrived at Lorentz’s transformation from a new direction and gave it a new perspective. There were no absolute  measures of space or time. A corollary was that Maxwell’s equations were the basic laws of the physical world. Newton’s laws were an approximation. From E=MC (squared), mass is just an immensely concentrated form of stored energy. All this based on the assumption that C is constant, and C is ratio  of electromagnetic and electrostatic units of charge. Einstein’s theory explained perfectly the way atomic particles behave when traveling close to the speed of light, and accounted for lost mass when a radioactive atom decays into two smaller ones. [xiii]

The combined theories of kinetic theory of gasses, thermodynamics, and Maxwell’s equations seemed to indicate that all of the kinetic energy of molecules should long ago have been radiated away, leaving a cold dead universe. What was missing? Nature seemed to have some hidden mechanism which allowed a balance to be reached between the radiation  that matter emitted and absorbed. Max Plank  produced a formula in an “act of desperation” which  allowed matter to absorb radiant energy only in discreet amounts, or quanta. Einstein completed the coup in 1905 by asserting that radiation itself comes in discreet packetd, now called “photons”. Maxwell made  the point that although the things we call photons and electrons appear to us to behave sometimes like particles and sometimes like waves, we should not make the mistake of thinking that they are either. [xiv]


[i] The Man Who Changed Everything: The Life of James Clerk Maxwell, by Basil Mahon. Wiley 2004.  p.56 f.


[ii] ibid p. 63 f.

[iii] ibid p. 95 f.

[iv] ibid p. 103.

[v] Ipid p. 105 f.

[vi] Ibid p. 120 f.

[vii] Ibid p. 200 f.

[viii] ibid p. 141 f.

[ix] ibid p. 147 f.

[x] ibid p. 162. f.

[xi] ibid p. 165

[xii] ibid p. 178 f.

[xiii] ibid p. 180 f.

[xiv] ibid p. 182 f.